The move for the Costa Rica international would see PSG’s French goalkeeper Alphonse Areola head the other way.
It would be a second spell in La Liga for Areola, who previously played for Villarreal.
Navas signed for Madrid from Levante in 2014 after impressing at the World Cup in Brazil.
The 32-year-old has made over 100 appearances for Real, winning three Champions Leagues and one La Liga crown. He has also lifted four FIFA Club World Cups, three UEFA Super Cups, and the Supercopa De Espana.
PSG could still be busy on Deadline Day with right-back Thomas Meunier tipped to join Juventus before the close of business in a swap deal that will see Mattia De Sciglio arrive at Parc des Princes.